S-Duality of Boundary Conditions In N=4 Super Yang-Mills Theory (0807.3720v1)

Abstract: By analyzing brane configurations in detail, and extracting general lessons, we develop methods for analyzing S-duality of supersymmetric boundary conditions in N=4 super Yang-Mills theory. In the process, we find that S-duality of boundary conditions is closely related to mirror symmetry of three-dimensional gauge theories, and we analyze the IR behavior of large classes of quiver gauge theories.

• The paper demonstrates a unified framework that links S-duality with boundary conditions via brane constructions and self-mirror theory T(G).
• The paper employs monopole operators to classify orthosymplectic quivers, revealing hidden symmetries and distinctions in IR behavior.
• The paper extends its analysis to U(n), SO(n), and Sp(n) groups, offering new perspectives on dual realizations in N=4 SYM.

Summary of "$S$-Duality of Boundary Conditions in $\mathcal{N}=4$ Super Yang-Mills Theory"

Gaiotto and Witten's investigation into boundary conditions in the $\mathcal{N}=4$ super Yang-Mills (SYM) theory is a pioneering work exploring the relationship between $S$-duality and boundary conditions in gauge theory, particularly focusing on unitary, orthogonal, and symplectic groups. This paper explores the deep correspondence between brane configurations in string theory and field-theoretical boundary conditions. By leveraging this relationship, the authors offer new insights into the nature of $S$-duality and its implications for boundary conditions within SYM theories.

The paper's central theme revolves around using brane constructions to analyze and understand $S$-dual boundary conditions. A powerful tool in their paper is the construction of the self-mirror theory $T(G)$ for a compact gauge group $G$, which plays a crucial role in understanding dualities. This theory displays $G \times G^\vee$ global symmetry and captures the essence of the $S$-dual of a Dirichlet boundary condition. By extending these ideas, Gaiotto and Witten develop a unified framework to describe the duality for any compact gauge group, including $U(n)$, $SO(n)$, and $Sp(n)$, each with their unique nuances and characteristics.

Significantly, the paper provides a comprehensive analysis of orthosymplectic quivers, extending earlier work on unitary quivers. The authors employ monopole operators to classify quivers as good, bad, or ugly based on the monopole operator's $R$-charge, revealing hidden symmetries and refining our understanding of the IR behavior of these systems. By identifying the parameters that determine whether an SCFT emerges or if monopole operators indicate a decoupled gauge sector, they extend the framework of three-dimensional mirror symmetry to new domains.

Key results in the paper include:

• The description of the $S$-dual of a given boundary condition in $G$ gauge theory, which involves embedding $T(G)$ or its variants, $T^\rho(G)$ and $T_{\rho^\vee}^\rho(G)$, into a holographic dual setup.
• An insightful resolution of the problem concerning which combinations of branes and orientifold or orbifold planes realize particular $S$-dual boundary conditions.
• Propositions on the infrared limits of orthosymplectic quivers, revealing how their Coulomb branches correlate with the symmetries of the system, and how utilizing notions from string theory influences their SCFT depiction.

The implications of these findings extend far and wide, suggesting new avenues for interpreting SYM theory boundary conditions while providing tools to solve outstanding problems concerning orthogonal and symplectic group dynamics in high-energy physics.

Gaiotto and Witten's methodology and findings offer a valuable resource for future research in mathematical physics and the paper of dualities across string theory and quantum field theories. While the theoretical constructs provided are built primarily using string-theory-inspired methodology, the potential for broader applications and generalizations across mathematics and quantum field theory represents a significant aspect of this work. Future developments in the use of quivers and monopole operators hold promise for new understandings and further expansion of the boundaries of theoretical physics.